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Pythagoras and trigonometry in 3D

3D problems are really 2D problems in disguise. Find the right-angled triangle hiding inside the shape.

The space diagonal

For a cuboid with sides a, b and c, the longest diagonal is √a2 + b2 + c2.
This is Pythagoras used twice: first across the base, then up.

Angles with a plane

To find the angle between a line and a base, draw the right-angled triangle formed by the line, its projection on the base, and the vertical height.

Method

1. Sketch the 3D shape.
2. Pick out a right-angled triangle and redraw it flat.
3. Use Pythagoras or SOHCAHTOA.
Worked example

A cuboid is 4 cm by 3 cm by 12 cm. Find the length of the space diagonal.

  1. Base diagonal: √16 + 9 = 5
  2. Space diagonal: √25 + 144 = √169

Answer: 13 cm

Key idea

Find the right-angled triangle inside the 3D shape and redraw it in 2D. Space diagonal = √(a2 + b2 + c2).

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